Short communicationLong short-term memory-based recurrent neural networks for nonlinear target tracking
Introduction
Target tracking is an important component in air space monitoring, video surveillance or navigation. Its role is to determine the number, position and motion parameters of targets from noisy observations [1]. When the detection procedure is involved, the implementing schemes can be grouped into two broad categories: the track-after-detection (TAD) and the track-before-detect (TBD) schemes [2], [3]. In this paper, the tracking algorithm is designed following the first scheme and the detection problem is not considered. For the nonlinear target tracking problem, estimation of the state of a system that changes over time is usually required using a sequence of noisy observations. In the Bayesian approach to dynamic state estimation, the posterior probability density function (PDF) of the state based on all available observations is constructed. Since this PDF embodies all available statistical information, an optimal estimate of the state may be obtained from it in principle [4]. However, the analytic solutions for the nonlinear target tracking problem are intractable to obtain because of the complex integrals except for a small number of particularly simple cases [5]. Instead, we usually resort to approximations or suboptimal solutions. The commonly used suboptimal methods include the extended Kalman filter (EKF), unscented Kalman filter (UKF) and particle filter (PF) [6].
EKF is proposed under the assumption that local linearizations of state transition and observation equations are sufficient descriptions of the nonlinearity. If the nonlinearity in these equations is very severe, its performance will be degraded significantly [6]. Motivated by which, UKF is proposed based on the statistical linearization rather than the analytical linearization like EKF. The UKF approximates the posterior by a Gaussian density, which is represented by a small number of deterministically chosen samples [6]. Therefore, it is not a global approximation due to very limited small set of trial points [6]. An alternative sampling-based method is the PF, which suffices to approximate complicated posteriors by using a large number of randomly chosen samples and has been successfully used in many areas [6]. Nevertheless, not only has it been suffering the sample degeneracy and impoverishment, but also the statistical effectiveness and computational efficiency are needed balancing in implementation [5], [7], [8].
Generally, target tracking problem can be regarded as a nonlinear mapping from an observation sequence to a target state vector. As a way to learn the nonlinear mapping based on automatic differentiation, the deep neural networks (DNNs) have made significant advances in fields such as image recognition, natural language processing and video processing [9], [10], [11]. Unlike most of traditional methods that are based on models, DNNs are developed by optimally learning the mathematical transformation from available training data [12]. Theoretically, a DNN is able to approximate any mapping to any precision given sufficient training data and suitable network structure [13], [14]. Therefore, it is expected that the relation between observations and true states of targets can be learned by a DNN, which is the main motivation of this paper. As a special DNN to handle sequential tasks, the recurrent neural networks (RNNs) are able to deterministically learn a conditional density of sequences by factorizing it and approximating each factor with a recurrent module, which coincides with the target tracking task in that the conditional density of target states given available observations is needed at each time step [15].
In [15], we proposed to use a vanilla RNN that is trained with least squares loss to address the motion uncertainty in target tracking where the dynamic models were linear. To handle the nonlinear target tracking problem, the RNN with long short-term memory (LSTM) structure is employed because of its superior performance in extracting features from sequential data [16]. Deep input-to-hidden and hidden-to-output structures are also used to allow the proposed networks to be more capable of fitting the mapping between observations and target states [17]. Furthermore, the proposed network in [15] was only capable of outputting the mean of posterior, which was a very limited description for the estimation of continuous values. To tackle this problem, a modified RNN-based structure with a modified loss is proposed that is capable of outputting both variance and mean of the estimation. Simulation results for an extensively used nonlinear time series model are presented, including the performance comparisons of the proposed RNN-based methods with the EKF, UKF and PF. It is shown that better estimation accuracy and shorter computational time can be obtained with the RNN-based methods. For the newly proposed RNN-based method, it is shown that the estimation error can be described by the output density properly. Even when the exact initial prior of the target state is considered unknown, both of the proposed RNN-based methods are able to provide nearly the same estimation accuracy.
Section snippets
Nonlinear target tracking
To define the problem of nonlinear target tracking, consider the evolution of state sequence of a target given bywhere is a process noise sequence and is a possibly nonlinear but known function of the target state with dimension nx and process noise with dimension nv; and are respectively the sets of natural numbers and real numbers. The objective of target tracking is to recursively estimate the target state xk from the observation zk
RNNs for nonlinear target tracking
An RNN can take as input a sequence by recursively processing input at each time step while maintaining its hidden state h [18]. At time step k, the RNN reads the input ik and updates its hidden state h aswhere is a deterministic nonlinear transition function and θh is its parameter set. For a vanilla RNN, the transition function only involves a nonlinear transformation of the sum of weighted input and hidden state [18]. The transition function can
Simulation results
Here, we consider a nonlinear time series model that has been used extensively in the literature for benchmarking numerical filtering techniques to demonstrate the improvement of performances [5], [23]. The state-space equations are as follows:where and and and ; denotes the Gaussian distribution with mean μ and variance σ2. The initial state distribution is . The representation in terms of
Conclusion
In this paper, two LSTM-based RNNs, namely, NRP-RNN and NRP-RNN-GO are proposed to address the nonlinear target tracking problem. The networks are designed according to the end-to-end principle by considering the filtering and prediction as separate tasks. Simulation results show that both of the RNNs are capable of providing more accurate estimations with shorter computational time compared to traditional model-based methods. Moreover, the NRP-RNN-GO is also able to provide an estimate of the
Declaration of Competing Interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work,there is no professional or other personal interest of any nature or kind in any product,service and/or company that could be construed as influencing the position presented in,or the review of, the manuscript entitled,”Long short-term memory-based recurrent neural networks for nonlinear target tracking”
Acknowledgments
This work was supported by the National Science Fund for Distinguished Young Scholars (61525105), the National Natural Science Foundation of China (61601340), the Fundamental Research Funds for the Central Universities (JB180215), the fund for Foreign Scholars in University Research and Teaching Programs (the 111 Project no. B18039).
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